The estimation of multivariate probability density functions has traditionally been carried out by mixtures of parametric densities or by kernel density estimators. Here we present a new nonparametric approach to this problem which is based on the integration of several multivariate histograms, computed over affine transformations of the training data. Our proposal belongs to the class of averaged histogram density estimators. The inherent discontinuities of the histograms are smoothed, while their low computational complexity is retained. We provide a formal proof of the convergence to the real probability density function as the number of training samples grows, and we demonstrate the performance of our approach when compared with a set of standard probability density estimators.